1- What do you think happened there? I didn't smell anything at the time.
Could be a number of things. Getting access to the internals and preferably a schematic would be a great help; without either, it's gambling more than anything.
I assume it's some kind of (fairly simple) triac circuit; in these, sometimes the triac fails, sometimes a driver/pulse generator, or it might be something as simple as a capacitor that has had its best time.
2- How can I disassemble the regulator?
I dunno, but the thing you're pointing at with the ballpoint looks like a rivet to me. Since these require access to both sides of the plate metal to install them, I suspect they don't have to be removed in order to open up the device. I bet it's there to hold an internal mounting bracket in place or something like that. You sure it's not some kind of snap-fit enclosure?
What can replace the regulator to connect to RH Designs F Stop timer?
Any ready-made solid state relay with a 110V trigger and input voltage rating and a power handling capacity that fits the enlarger (unlikely to be an issue with most solid state relays). I've never looked for them, but given the omnipresence of solid state relays these days, I'm sure there's some nicely boxed units around that can act as a snap-in replacement.